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Influence of Vertical Vibrations on the Stability of a Binary Mixture in a Horizontal Porous Layer Subjected to a Vertical Heat Flux

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Abstract

We present an analytical and numerical stability analysis of Soret-driven convection in a porous cavity saturated by a binary fluid mixture and subjected to vertical high-frequency and small-amplitude vibrations. Two configurations have been considered and compared: an infinite horizontal layer and a bounded domain with a large aspect ratio. In both cases, the initial temperature gradient is produced by a constant uniform heat flux applied on the horizontal boundaries. A formulation using time-averaged equations is used. The linear stability of the equilibrium solution is carried out for various Soret separation ratios \(\varphi \), vibrational Rayleigh numbers Rv, Lewis numbers Le and normalized porosity. For an infinite horizontal layer, the critical Rayleigh number \(\mathrm{Ra}_c\) is determined analytically. For a steady bifurcation to a one-cell solution (the critical wavenumber is zero), we obtain \(\mathrm{Ra}_{c}={12}/{({\varphi }(\mathrm{Le}+1)+1)}\) for all Rv. When the bifurcation is a Hopf bifurcation or when the critical wavenumber is not zero, we use a Galerkin method to compute the critical values. Our study is completed by a nonlinear analysis of the bifurcation to one-cell solutions in an infinite horizontal layer that is compared to numerical simulations in bounded horizontal domains with large aspect ratio.

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Acknowledgements

The authors are grateful to CNES (Centre National d’Etudes Spatiales) for its financial support.

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Authors and Affiliations

  1. Institut de Mécanique des Fluides de Toulouse, IMFT, Université de Toulouse, CNRS, Toulouse, France

    Soumaya Ouadhani, Abdelkader Mojtabi & Alain Bergeon

  2. Mathematical Engineering Laboratory, Tunisia Polytechnic School, University of Carthage, El Khawerzmi street, B.P.743, 2078, La Marsa, Tunisia

    Soumaya Ouadhani & Ali Abdennadher

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  1. Soumaya Ouadhani
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  2. Ali Abdennadher
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  3. Abdelkader Mojtabi
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  4. Alain Bergeon
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Correspondence to Alain Bergeon.

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Ouadhani, S., Abdennadher, A., Mojtabi, A. et al. Influence of Vertical Vibrations on the Stability of a Binary Mixture in a Horizontal Porous Layer Subjected to a Vertical Heat Flux. Transp Porous Med 124, 203–220 (2018). https://doi.org/10.1007/s11242-018-1059-5

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  • DOI: https://doi.org/10.1007/s11242-018-1059-5

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